From 2e32bffea8ab09762cffc8556361121c0aaea33a Mon Sep 17 00:00:00 2001 From: egorzhurov Date: Mon, 13 May 2024 16:54:09 +0300 Subject: [PATCH] Move 42 article to markdown format --- lessons/java-core/042/Tree.md | 444 ++++++++++++++++++++++++++++ lessons/java-core/042/picture1.png | Bin 0 -> 46279 bytes lessons/java-core/042/picture10.png | Bin 0 -> 24997 bytes lessons/java-core/042/picture11.png | Bin 0 -> 36295 bytes lessons/java-core/042/picture12.png | Bin 0 -> 36217 bytes lessons/java-core/042/picture13.png | Bin 0 -> 35682 bytes lessons/java-core/042/picture14.png | Bin 0 -> 32233 bytes lessons/java-core/042/picture2.png | Bin 0 -> 64289 bytes lessons/java-core/042/picture3.png | Bin 0 -> 35688 bytes lessons/java-core/042/picture4.png | Bin 0 -> 354471 bytes lessons/java-core/042/picture5.png | Bin 0 -> 239794 bytes lessons/java-core/042/picture6.png | Bin 0 -> 276058 bytes lessons/java-core/042/picture7.png | Bin 0 -> 283657 bytes lessons/java-core/042/picture8.png | Bin 0 -> 193567 bytes lessons/java-core/042/picture9.png | Bin 0 -> 309679 bytes 15 files changed, 444 insertions(+) create mode 100644 lessons/java-core/042/Tree.md create mode 100644 lessons/java-core/042/picture1.png create mode 100644 lessons/java-core/042/picture10.png create mode 100644 lessons/java-core/042/picture11.png create mode 100644 lessons/java-core/042/picture12.png create mode 100644 lessons/java-core/042/picture13.png create mode 100644 lessons/java-core/042/picture14.png create mode 100644 lessons/java-core/042/picture2.png create mode 100644 lessons/java-core/042/picture3.png create mode 100644 lessons/java-core/042/picture4.png create mode 100644 lessons/java-core/042/picture5.png create mode 100644 lessons/java-core/042/picture6.png create mode 100644 lessons/java-core/042/picture7.png create mode 100644 lessons/java-core/042/picture8.png create mode 100644 lessons/java-core/042/picture9.png diff --git a/lessons/java-core/042/Tree.md b/lessons/java-core/042/Tree.md new file mode 100644 index 0000000..82569a9 --- /dev/null +++ b/lessons/java-core/042/Tree.md @@ -0,0 +1,444 @@ +# Структура данных Дерево + +В рамках предыдущих уроков мы познакомились с некоторыми структурами данных: массив, несколько видов связанного списка, +стек и несколько видов очередей. Каждая из этих структур имеет свои характерные особенности, но все они относятся к +одной большой группе – **линейные структуры данных**. + +Линейные структуры характеризуются тем, что каждый элемент имеет не более одного предка и не более одного потомка +(предыдущего и следующего элементов соответственно). + +Однако структуры данных в массе своей куда более разнообразны. Сегодня мы познакомимся с большой группой **нелинейных +структур данных – деревьями**. Нет, не хвойными и не плодовыми. Мичуринцев из вас не получится:) + +Именно на базе деревьев работают некоторые из коллекций в Java. Например, знакомый нам по предыдущему уроку `TreeSet`. + +В общем случае, дерево – структура данных, у каждого из узлов которой есть не более одного предка, и может быть любое +количество потомков. Данное определение нельзя назвать академическим, но, на мой взгляд, оно отображает суть данной +группы структур. + +Примеры визуализации деревьев можно увидеть ниже: + +![img.png](picture1.png) + +Именно возможность наличия у одного узла нескольких «следующих» узлов и делает подобную структуру нелинейной. + +В целом, древовидные структуры существуют и в реальной жизни. Например, иерархия подчинения сотрудников в компании +представляется классической древовидной структурой. Структура папок в файловой системе компьютера тоже может быть +представлена в виде дерева. + +Но при разработке ПО недостаточно интуитивного понимания древовидных структур. Именно поэтому сегодня мы познакомимся с +ними поглубже. + +> !NB: Объекты в хипе (куче) – одной из областей памяти в JVM – тоже хранятся в структуре типа дерево. + +## Основные термины + +Поскольку деревья являются намного более сложной структурой, нежели изученные нами представители линейных структур, +предлагаю познакомиться с терминологией, применимой при описании и обработке деревьев. «Первый» и «последний» элементы +уже явно недостаточные термины для описания подобной структуры. + +**Узлы** дерева (вне зависимости от их расположения) могут называться **вершинами**. + +**Корень дерева (корневой узел, root)** – узел дерева, не имеющий предка. Является **точкой входа** в дерево, так же, +как **вершина** – точка входа в стек (не путать с вершиной в контексте деревьев). + +**Лист (терминальный узел, leaf)** – узел дерева, не имеющий потомков. Если корень у дерева один, то листьев, как +правило, много. + +Также иногда можно встретить термин **«внутренний узел»** – любой узел дерева, не являющийся корнем или листом +(соответственно, имеющий и предка, и потомка(-ов)). + +Теперь, глядя на Рисунок 1, мы можем определить, что у дерева б есть: + +1. Корень (узел А); +2. Внутренние узлы (B, C, E, F); +3. Листья (D, G, H, J). + +Также в случае с деревьями возникает понятие **«глубины дерева»**. Поскольку структура данных не является линейной, путь +от корня до другого узла слабо коррелирует с количеством элементов в структуре. + +![img.png](picture2.png) + +Зато коррелирует с количеством **уровней** в дереве. + +Итак: + +**Уровень дерева** – абстракция для группы узлов дерева, указывающая на количество узлов от корня до любого узла из +группы. + +На Рисунке 2 дана визуализация уровней. Корень находится на нулевом уровне, элементы, являющиеся потомками (их также +называют **ветвями**) корня – находятся на первом уровне, потомки узлов первого уровня – на втором и т.д. + +**Глубина (высота) дерева** – количество уровней в дереве. Например, на Рисунке 2 изображено дерево с глубиной 4 (не +забывайте, true-программисты считают с нуля). + +По сути, глубина дерева – это количество итераций перехода между элементами, необходимое, чтобы добраться от корня до +самого дальнего листа (как можете увидеть на Рисунке 2, листья могут находиться на разных уровнях). + +И, наконец, познакомимся с термином **«поддерево»**: поддеревом называется дерево, являющееся частью другого дерева. + +Например, на Рисунке 2 можно выделить поддерево от вершины _«E»_, в таком случае дерево будет иметь 3 элемента: корень +_«Е»_ и листья _«H»_ и _«J»_. + +Также мы можем выделить и поддерево с корнем _«C»_ - такое поддерево будет иметь лишь одного потомка и, де-факто, +выродится в связный список из двух элементов (_«C»_ и _«F»_). + +## Виды деревьев + +Деревья бывают разные, поэтому внутри этой группы структур есть своя классификация (и не одна). Она достаточно обширна, +в рамках урока мы познакомимся лишь с теми видами и представителями, которые нам могут понадобиться в обозримом будущем. + +С точки зрения типа деревьев, в рамках урока нас интересуют, в первую очередь, **бинарные (двоичные) деревья**. Не +путайте их с **b-tree (В-деревьями)**. С последними советую познакомиться самостоятельно, если текущий урок дался вам +легко, гиперссылка в предыдущем предложении. + +Итак, бинарные деревья – деревья, в которых каждый узел имеет не более двух потомков. При реализации бинарных деревьев в +программном коде, потомков узла обычно так и обозначают: _left_ и _right_. Ниже можно увидеть пример двоичного дерева: + +![img.png](picture3.png) + +Также для деревьев (не только бинарных, но для нас это не важно) существует понятие **балансировки**. + +В общем случае, бинарное дерево можно считать **сбалансированным**, если для любого узла верно, что количество элементов +в его левом и правом поддеревьях, отличается не более, чем на единицу. Такое определение описывает идеальную +балансировку дерева. + +Менее строгим будет определение для балансировки по высоте: дерево является **сбалансированным (по высоте)**, если +расстояние пути от корня до любого из листьев отличаются не более, чем на единицу. Иными словами, все листы (элементы +без потомков) расположены в рамках одного или двух соседних уровней. Например, дерево на Рисунке 3 является +сбалансированным по высоте. Но применяя более общее определение балансировки, то же дерево назвать сбалансированным уже +нельзя. + +Балансировка дерева необходима для того, чтобы поиск по дереву был максимально эффективен. Если же балансировку не +производить – вполне вероятны сценарии, когда дерево (или, чаще, ряд поддеревьев) вырождаются в списки или глубина +левого и правого поддерева какого-то из узлов оказывается сильно различной, соответственно, эффективность доступа к +листьям различных поддеревьев будет сильно различаться. + +Полагаю, достаточно очевидно, что при балансировке у дерева может измениться корень – логично, что корнем +сбалансированного дерева должно быть значение, близкое к медианному для элементов дерева. Иными словами, если выписать +элементы дерева в отсортированном виде, корневым, в идеале, стоит делать элемент, находящийся в середине списка. Чтобы +количество элементов слева от выбранного, и справа, было равно (или почти равно). + +На практике, не всегда корнем сбалансированного дерева будет медианный элемент – операция балансировки является +достаточно дорогой при любой реализации, вызывать ее после каждой операции вставки/удаления элемента – крайне +сомнительная затея. + +В качестве частного случая двоичного дерева предлагаю рассмотреть **двоичное дерево поиска**. + +_Примечание_: если хочется нырнуть глубже в тему деревьев – предлагаю после освоения этого урока самостоятельно +ознакомиться с понятием **двоичной кучи**. Вряд ли эти знания пригодятся в рамках текущего курса, но точно будут +полезны, если вы более-менее уверенно чувствуете себя в структурах данных. + +Итак, двоичным деревом поиска называется двоичное дерево, которое удовлетворяет следующим условиям: + +- Для любого узла верно, что его левое поддерево (поддерево, где корнем взят левый потоком текущего узла) содержит лишь + меньшие или равные значения; +- Для любого узла верно, что его правое поддерево содержит лишь большие значения; +- Для любого узла верно, что оба его поддерева (левое и правое) являются двоичными деревьями поиска (читай, являются + двоичными деревьями и выполняют оба условия выше). + +Как видите, описание условий для дерева сильно напоминает рекурсивные. В общем-то, обработка деревьев – одна из +классических областей применения для рекурсии. + +К слову, Рисунок 3 демонстрирует именно двоичное дерево поиска. + +Различные реализации такого типа деревьев отличаются, в первую очередь, условием, при котором будет запущена +балансировка. С этим мы познакомимся чуть позже. + +Но, чтобы понять, как работает балансировка дерева (спойлер: по-разному, алгоритмы балансировки могут отличаться даже +для одной и той же реализации дерева может существовать несколько алгоритмов балансировки для разных ситуаций), стоит +сначала разобраться с более простыми операциями: вставки и удаления. + +## Добавление элемента в двоичное дерево поиска + +Вставка в двоичное дерево поиска происходит следующим образом: + +1. Если дерево пустое (элементы отсутствуют), сделать добавляемый элемент корневым; +2. В противном случае сравнить элемент с корнем: если он больше текущего значения – вызвать функцию добавления для + поддерева, где корнем будет правый потомок текущего узла, в иных случаях `<=` – вызвать функцию добавления в + поддерево с корнем в лице левого потомка текущего узла; +3. Через какое-то количество рекурсивных вызовов шага 2, функция добавления будет вызвана для пустого поддерева, что, в + свою очередь, равносильно шагу 1. + +Если попытаться визуализировать алгоритм добавления, он сведется к сравнению добавляемого элемента с текущими и +переходом к следующей итерации сравнения (с правым или левым потомком) до тех пор, пока добавляемый элемент не будет +сравнен с `null` (отсутствующим значением). Именно на место отсутствующего значения и будет вставлен элемент. + +Функция добавления элементов достаточно проста в реализации, но, как и большинство рекурсивных алгоритмов, достаточно +тяжела для описания. + +## Удаление элемента из двоичного дерева поиска + +Функция удаления может работать по-разному, в зависимости от конкретной реализации. Предлагаю рассмотреть один из +возможных вариантов: + +1. Найти удаляемый элемент; +2. Если элемент отсутствует (включая ситуацию, когда дерево пустое) – закончить выполнение, иначе перейти к п.3; +3. Если потомков нет (удаляемый элемент является листом дерева) – заменить ссылку на удаляемый элемент в родительском + ссылкой на `null`, иначе – перейти к п.5; +4. Если удаляемый элемент имеет лишь одного потомка (правую или левую ветвь) – присвоить предку удаляемого элемента + ссылку на потомка (вместо ссылки на удаляемый элемент) для соответствующей ветви. Этот пункт напоминает удаление + элемента из середины односвязного списка. Если потомков 2 – перейти к п.5; +5. Найти среди левого поддерева наибольший элемент и перенести его на место текущего (для найденного элемента будет + применен п.3). + +Таким образом, ссылка на удаляемый элемент исчезнет из иерархии дерева, вместо нее будет `null` или ссылка на другой +элемент. Обратите внимание, если дерево может содержать лишь уникальные элементы, в п.5 может быть как наибольший +элемент из левого поддерева, так и наименьший из правого – в обоих случаях это будет элемент, максимально близкий к +удаляемому по значению. В нашем случае выбран вариант с поиском в левой ветви, потому что в ней могут содержаться +дубликаты удаляемого значения (полные, или имеющий одинаковый вес для выбранного в данном дереве параметра сортировки – +не имеет значения). + +Также стоит обратить внимание, что в классической реализации, если дерево будет содержать несколько одинаковых +элементов, то при попытке удалить указанный элемент будет удален лишь один (наиболее близкий к корню), а не все. + +Теперь, после знакомства с операциями добавления и вставки, предлагаю разобрать несколько известных реализаций двоичного +дерева поиска и нюансы балансировки, которые для них характерны. + +## АВЛ-дерево + +**АВЛ-дерево** – двоичное дерево поиска, сбалансированное по высоте (выше мы разобрались, что это значит). + +Удаление элемента в данном виде дерева происходит по принципу, описанном в предыдущем пункте статьи. + +Хорошая новость заключается в том, что в рамках данного определения все очень легко. Плохая – в том, что вся суть +АВЛ-дерева в способах его балансировки и именно с ними мы познакомимся ниже (да, их несколько). + +Балансировка АВЛ-дерева происходит после вставки нового значения, если эта вставка привела к тому, что разница высот +правого и левого поддерева любого из узлов стала равна 2. Напомню, при балансировке по высоте допустимая разница – 0 +(высоты равны) или 1. + +Всего существует 4 способа балансировки АВЛ-дерева, каждый из которых используется для своих условий. + +Ниже будут использованы слайды презентации, в которой, на мой взгляд, наиболее доступно и лаконично описаны способы +балансировки АВЛ-дерева. Если кому-то зашел формат описания, можете ознакомиться с +[презентацией](https://ppt-online.org/13500) целиком. Она, посвящена, преимущественно, разбору АВЛ-дерева и дает более +исчерпывающее описание и примеры по данной структуре данных. + +### Малое правое вращение + +![img.png](picture4.png) + +![img.png](picture5.png) + +Обратите внимание на способ визуализации узлов и поддеревьев, в т.ч. и то, что визуально из одного узла может выходить +узел (с поддеревьями) и поддерево одновременно (на первом слайде это также подчеркнуто). Такая форма визуализации очень +лаконична, но непривычна и тяжела для восприятия новичками. Тем не менее, именно она используется повсеместно (если +захотите глубоко нырнуть в тему деревьев – стоит привыкать уже сейчас). + +### Малое левое вращение + +![img.png](picture6.png) + +В целом, симметрично малому правому. + +### Большое правое вращение + +Небольшая расшифровка легенды перед слайдом: + +- _TX_ – дерево (поддерево) с корнем в узле _X_; +- _ТА_ – дерево с корнем в узле _А_; +- _ТС_ – дерево с корнем в узле _С_. Оно же _LR_ (на левом изображении); +- _Т2_ и _Т3_ – левое и правое поддеревья С соответственно. Т.е. _С_ – предок для корней поддеревьев _Т2_ и _Т3_. + +![img.png](picture7.png) + +![img.png](picture8.png) + +### Больше левое вращение + +![img.png](picture9.png) + +Опять же, симметрично большому правому вращению. + +Для закрепления понимания принципов работы АВЛ-дерева рекомендую ознакомиться со следующей +[ссылкой](https://www.cs.usfca.edu/~galles/visualization/AVLtree.html) + +Она позволяет посмотреть визуализацию добавления и удаления элементов пошагово. Переключение шагов доступно в +автоматическом и ручном режимах. + +## Красно-черное дерево + +Если АВЛ-дерево является очень чувствительным к вставкам, результатом чего является частая перебалансировка, то +следующий рассматриваемый нами вид деревьев, является менее прихотливым, хоть и все еще самобалансирующимся. + +Итак, **красно-черное дерево**. Оно же **red-black tree**, оно же **RB-tree** (именно такое название вы будете слышать +чаще всего в профессиональной деятельности). + +![img.png](picture10.png) + +Характерной особенностью этого дерева является дополнительный параметр узла – его _«цвет»_. Этот параметр бывает двух +видов: красный или (неожиданно) черный. От чего это зависит мы разберем ниже. + +Итак, особенности RB-tree: + +1. Каждый узел имеет цвет и двух потомков; +2. Листья – всегда фиктивные, т.е. узел есть, а данных в нем (полезной нагрузки) – нет. Именно поэтому каждый + узел [с заполненным значением] имеет двух потомков. На самом деле, есть реализации, не выполняющие этого условия, но + это не входит в рамки сегодняшнего урока; +3. Листья – всегда черные; +4. Корень – всегда черный (на самом деле, не имеет значения, просто традиция); +5. Потомки любого красного узла – черные; +6. Пусть от любого узла до любого из листьев-потомков должен иметь равное число черных узлов. + +В описанных условиях получается, что наличие красных узлов – вообще не обязательно. На самом деле, это не совсем так С +учетом п.6. – совершенно не так, если дерево содержит четное количество элементов, т.е. каждую вторую вставку:) + +Красно-черное дерево является более сложным в контексте операций вставок и удаления из-за необходимости проверки цвета +узлов и потенциального перекрашивания. В рамках урока кажется избыточным разбирать это в подробностях – на первых порах +хватит разбора на примере АВЛ-дерева. + +Отметим лишь ряд особенностей, характерных для красно-черного дерева: + +- Добавляемый в дерево узел изначально маркируется красным. В дальнейшем его цвет может быть изменен на черный в + зависимости от цвета близлежащих узлов. Этот момент подсвечен исключительно для понимания, откуда вообще в + красно-черном дереве берутся красные узлы; +- RB-tree, в отличии от АВЛ-дерева, балансируется не при разности высот >1, а в ситуации, когда количество черных узлов + в пути от вершины до листьев становится различным. Т.е., на практике возможна ситуация, когда путь до крайнего (не + считая пустые листовые) потомка по одной ветви в два раза короче, чем путь до потомка по другой. Потому что одна ветвь + содержит лишь черные узлы, а вторая – чередует красные и черные (напомню, два красных узла подряд быть не может). Зато + операция перекрашивания узлов, в отличии от перебалансировки, работает почти на каждое добавление; +- За счет особенности условий перебалансировки из предыдущего пункта, RB-tree, безусловно, менее эффективно, чем + АВЛ-дерево при поиске, но разница в эффективности находится в пределах 39%. Обычно она еще меньше; +- На практике, чаще всего для хранения данных используется именно красно-черное дерево, а не АВЛ. Да, оно, зачастую, + чуть менее эффективно при поиске, но добавление и удаление элементов там, зачастую, дешевле за счет менее строгих + требований к балансировке. В том числе, на базе RB-tree работает ряд коллекций в Java (об этом в следующем уроке). + +Как и в предыдущем пункте, вы можете посмотреть на +[визуализацию операций](https://www.cs.usfca.edu/~galles/visualization/RedBlack.html) в красно-черном дереве: + +## Обход дерева + +**Обход дерева** – термин, обозначающий поиск элемента в дереве, при котором каждый узел «посещается» лишь 1 раз. Также +называется **«поиском по дереву»**. + +В данном случае, под поиском подразумевается не нахождение конкретного элемента в дереве поиска (это можно сделать +намного эффективнее, чем проверить все узлы дерева), а выборка элементов вне дерева поиска (или в нем, но по признаку, +отличному от того, на основании которого распределялись элементы в дереве). Также может иметь смысл, если необходимо +выбрать все элементы, удовлетворяющие критерию поиска. + +Кроме, непосредственно, поиска, актуально для любых операций, которые необходимо сделать со всеми элементами дерева. + +В рамках статьи мы коснемся лишь обхода бинарного дерева. + +До того, как перейдем к конкретным алгоритмам обхода, отмечу, что дерево можно обходить (безотносительно конкретного +алгоритма обхода) как слева-направо, так и справа-налево. Это не имеет серьезного влияния на алгоритм, поэтому при +разборе ниже этот фактор опускается. На практике, обычно реализуют обход слева-направо. + +## Обход в глубину + +**Обход в глубину** – общее название для алгоритмов обхода дерева, подразумевающих полный обход каждого поддерева до +перехода к следующему поддереву. Его альтернативой является обход в ширину, который мы кратко затронем чуть ниже. + +**Прямой обход** – алгоритм обхода дерева, который может быть описан следующей последовательностью шагов +(п.2 – само действие с элементов, пп.3-4 – рекурсивные вызовы): + +1. Проверка узла на `null`; +2. Действие с узлом (проверка условия поиска или др.); +3. Вызов функции прямого обхода дерева для левого поддерева; +4. Вызов функции прямого обхода дерева для правого поддерева. + +Визуализация порядка обработки элементов при прямом обходе. Точками обозначено «действие с узлом», с учетом порядка +обработки элементов (пунктирная линия): + +![img.png](picture11.png) + +**Центрированный обход** – алгоритм, описываемый следующими шагами +(п.3 – само действие с элементов, пп.2,4 – рекурсивные вызовы): + +1. Проверка узла на `null`; +2. Вызов функции центрированного обхода дерева для левого поддерева; +3. Действие с узлом (проверка условия поиска или др.); +4. Вызов функции центрированного обхода дерева для правого поддерева. + +Визуализация: + +![img.png](picture12.png) + +**Обратный обход** может быть описан с помощью следующей последовательности шагов +(п.4 – само действие с элементов, пп.2-3 – рекурсивные вызовы): + +1. Проверка узла на `null`; +2. Вызов функции обратного обхода дерева для левого поддерева; +3. Вызов функции обратного обхода дерева для правого поддерева; +4. Действие с узлом (проверка условия поиска или др.). + +Визуализация: + +![img.png](picture13.png) + +## Обход в ширину + +Обход в ширину является принципиально другим подходом. Она заключается в том, чтобы сначала обработать все элементы на +одном уровне, а лишь потом переходить к следующему. + +Визуально обход в ширину можно представить следующим образом: + +![img.png](picture14.png) + +Важной особенностью такого подхода является необходимость хранения элементов, обработка которых откладывается. Если при +обходе в глубину хранение «отложенного» элемента можно делегировать стеку рекурсивных вызовов (альтернативно, можно +сохранять в стеке явно, но не суть), то при обходе в ширину явно сохранять отложенные элементы придется в любом случае. +Обычно для этого используются очереди (сейчас речь именно о структуре данных, а не о типе Java-коллекций). + +Таким образом, обход в ширину является менее очевидным алгоритмом, хотя, в сущности, остается несложным. Но иногда он +может быть полезным. Например, визуализировать дерево в консоли с помощью псевдографики определенно удобнее, используя +обход в ширину. + +## Итог + +Тема деревьев является очень обширной и интересной. Мы затронули ее совсем поверхностно, на уровне, чуть более глубоком, +чем необходимо для дальнейшего изучения Java-коллекций. На самом деле, средний разработчик (не только джуниор или миддл, +но, подозреваю, и многие, из называющих себя сеньорами), скорее всего знает не больше (вероятно, даже меньше), чем +описано в данной статье. Поэтому нет ничего страшного, если на данном этапе вы ограничитесь этим материалом, при +условии, что смогли в него вникнуть и понять в полном объеме. Однако я надеюсь, что кому-то станет интересно +познакомиться с темой глубже. Для того, чтобы чуть облегчить погружение, в статье есть несколько ссылок, еще пару штук я +прикреплю ниже. + +Отдельное спасибо хочу выразить коллеге, который помог с материалами для статьи. Именно благодаря ему в статье появились +ссылки на интерактивные визуализации работы АВЛ-дерева и RB-tree, а также были дополнены некоторые пункты и расширен +список ссылок для ознакомления. + +## Собственно, ссылки + +Альтернативная [статья](https://otus.ru/journal/derevo-kak-struktura-dannyh/) для первого знакомства + +Тяжелая артиллерия при знакомстве со структурами данных: **Никлаус Вирт. Алгоритмы и структуры данных** (легко гуглится) + +Подробное разжевывание красно-черных деревьев в двух частях. [Ссылка на первую](https://habr.com/ru/post/555404/) + +Также для заинтересовавшихся рекомендую ознакомиться с **B-деревьями** (ссылка была в статье), структурами, производными +от **B-tree (B*-tree, B+-tree)**, а также 2-3-деревьями (отсылка на них была в предыдущей ссылке). Как минимум B-tree +деревья понадобятся для более глубокого понимания работы индексов в БД (правда, нам до этого еще далеко). + +#### С теорией на сегодня все! + +![img.png](../../../commonmedia/defaultFooter.jpg) + +Переходим к практике: + +## Задача 1: + +Реализуйте бинарное дерево поиска. Учтите возможность использования дерева как для _Comparable_-сущностей, так и для +сортировки на основе компаратора. + +Реализуйте в рамках дерева методы, производящие обход в глубину (любой из трех рассмотренных в статье) и ширину. +Результатом работы этих методов должно быть перечисление элементов дерева в консоли в порядке их обхода. + +Обратите внимание: в этой задаче не идет речи об автобалансировке дерева. + +Опциональное условие(*): реализуйте метод балансировки дерева по высоте. В рамках текущей задачи принимается реализация +любой степени наивности, если она дает корректный результат. + +## Задача 2 (**): + +Реализуйте АВЛ-дерево. Сопроводите действия по добавлению (и последующей балансировке) и удалению элементов выводом +соответствующих сообщений в консоль. + +> Если что-то непонятно или не получается – welcome в комменты к посту или в лс:) +> +> Канал: https://t.me/ViamSupervadetVadens +> +> Мой тг: https://t.me/ironicMotherfucker +> +> **Дорогу осилит идущий!** \ No newline at end of file diff --git a/lessons/java-core/042/picture1.png b/lessons/java-core/042/picture1.png new file mode 100644 index 0000000000000000000000000000000000000000..e97baa4bbd01ccdd5cc45a36aa9e052d1e90ab1f GIT binary patch literal 46279 zcmXt91yoeg)>i56?jfbS8-`Bl?ohhByG5lN6r~xuyCfu}ySsDf|N7qhS;HD;)?Jr# z_t|I1_kBA?O+^+BnHc%an>T3ka?&5(ym<$C^X9D%B0TU-JsUn7@b=d2gRJD6%0HwB zzz0|x31x{lZy-r1PiAnyXC!AiJ-0V+(0l*=z8!S>W%=e!ouRz6gr>LgaSoiWrXtDU znbyH6Ivgw$`%;R}~027Hy`Yu33MDzAS zVt%R797945J`AC@;o(7O_ShJBE-r4TRbmRUAp8ngDzg7y zRguozsk3`sROhR}BuK~~gM2&0{EA=|{^Wy+9bxg|gNFF&1F&5SDEPMZA0K6YZU5~mwK4#g9UrW=^G#1{ 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